Measurement of Dynamic Proppant Fall Rates in Fracturing Gels Using a Concentric Cylinder Tester (includes associated paper 9970 )

Abstract

Summary This paper describes the measurement of proppant terminal velocity in a simple non-Newtonian gel during shear. The shear was imposed between Lucite™cylinders with the outer cylinder rotating. During rotation, a proppant particle was introduced and the terminal velocity was measured. This measurement then was compared with Stokes law measurement then was compared with Stokes law using the fluid's apparent viscosity at the known shear rate for the Newtonian viscosity. This results in a good correlation between measured and theoretical data. Introduction Most of the early work done on sand settling used vertical fracturing models. Kern et al.1 and Babcock et al.2 reported on tests performed in such models which defined equilibrium bank height and equilibrium velocities. Novotny3also used a vertical model in his proppant transport study. Wahl4 studied the transport of proppants in a horizontal model. A knowledge of particle settling velocity is a necessary input in all design techniques that describe the final location of proppants in the fracture. For Newtonian fluids, the measurement of settling velocity is straightforward since their viscosity remains constant with variations in shear rate. For these fluids, particle settling velocity will follow Stokes law for particle Reynolds numbers less than two. For all but very thin fracturing fluids, particle Reynolds numbers will be in this range. Under these conditions, single particle settling velocity will follow Stokes law:Equation 1 This is the settling velocity for a single particle in an infinite media. Most fracturing gels, on the other hand, are non-Newtonian in character and are considered to follow the power law:Equation 2 Under these conditions, Stokes law becomesEquation 3 From this equation, settling velocity becomes a function of shear rate. As shown by Novotny,3 shear rate (?) consists of two components: a horizontal component imposed by fluid motion and a vertical component imposed by particle settling. The vertical shear rate can be expressed by V/d, and the combined shear rate becomesEquation 4 This shear rate expression leads to an altered form of Eq. 3. For the fluids used in these tests, the vertical shear rate V/d is very small and can be ignored without significant error. The experiments conducted were designed to see how closely dynamic particle fall velocity would follow Stokes law for the shear rate range imposed by the test device. If the measured values agreed with Stokes law, only a knowledge of shear rate would be required to predict settling velocity. Novotny3 found a good correlation working with a similar tester. The fluids used in our test were somewhat more viscous than shown by Novotny.3

Zero Shear Viscosity Determination of Fracturing Fluids: An Essential Parameter In Proppant Transport Characterizations

“…8 The work of Hannah and Harrington was prominently referenced. 9 They reported that the settling rate was relatively independent of the shear rate above 20 sec -1 shear rates.…”

New model for single spherical particle settling velocity in power law (visco-inelastic) fluids

“…Subsequently, Hannah and Harrington (1981) demonstrated that it was not possible to predict the settling velocity simply from knowledge of the shear rate. Shah (1982Shah ( , 1986 presented a different approach to analyze the particle settling velocity data in non-Newtonian fluids.…”

Micromechanics of proppant agglomeration during settling in hydraulic fractures

“…Curve-fitting models were previously derived for predicting proppant settling rates, including the effects of the particle concentration in the fluid and the relative size of the fracture width relative to the proppant diameter. For proppant settling in pipes and narrow slots, existing experimental correlations relate average slurry settling velocity to a single particle settling velocity based on particle concentrations assuming uniform spatial distribution in the fluid (Hannah and Harrington 1981;Harrington et al 1979;Novotny 1977;Shah 1982Shah , 1993Shah and Lee 1986;Shah and Lord 1990;Shah et al 2007). Particle agglomerations and irregular particle motions were previously observed in several experimental studies on flow of particle-fluid mixtures in pipes and slot flow apparatus.…”